Results 131 to 140 of about 2,769 (160)

Hausdorff and packing measure for solenoids

Ergodic Theory and Dynamical Systems, 2003
Summary: We prove that the solenoid with two different contraction coefficients has zero Hausdorff and positive packing measure in its own dimension and the SBR measure is equivalent to the packing measure on the attractor. Further, we prove similar statements for Slanting Baker maps with intersecting cylinders (in \(\mathbb{R}^{2}\)).
Rams, Michał, Simon, Károly
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Hausdorff and Packing Measures of Compactly Nonrecurrent Regular Elliptic Functions

2023
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Hausdorff and Packing Measures

Fractals and Dynamics in Mathematics, Science and the Arts
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Scaling properties of Hausdorff and packing measures

Mathematische Annalen, 2001
Let \(\theta \) be a continuous increasing function defined on the nonnegative number line with some restriction. Among other results, the authors characterize those function \(\theta \) such that the corresponding Hausdorff or packing measure with gauge function \(\theta \) scales with exponent \(\alpha \) by showing it must be a product of a power ...
Csörnyei, Marianna, Mauldin, R. Daniel
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Hausdorff and packing dimensions of Mandelbrot measure

International Journal of Mathematics, 2020
We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text].
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Hausdorff and packing dimensions and sections of measures

Mathematika, 1998
Summary: Let \(m\) and \(n\) be integers with \(0< m< n\) and let \(\mu\) be a Radon measure on \(\mathbb{R}^n\) with compact support. For the Hausdorff dimension, \(\dim_H\), of sections of measures we have the following equality: for almost all \((n- m)\)-dimensional linear subspaces \(V\) \[ \text{ess inf}\{\dim_H \mu_{V,a}: a\in V^{\perp}\text ...
Järvenpää, Maarit, Mattila, Pertti
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Self-similar sets of zero Hausdorff measure and positive packing measure

Israel Journal of Mathematics, 2000
The authors prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension. For instance, if \(1/5< r< 1/3\) then the set \({\mathcal K}^r_u\) of all sums \(\sum^\infty_{n=0} a_nr^n\) with \(a_n\in \{0, 1,u\}\) has this property for almost every \(u\) from a certain nonempty interval ...
Peres, Yuval, Simon, Károly, Solomyak, Boris   +2 more
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Hausdorff and packing measure functions of self-similar sets: continuity and measurability

Ergodic Theory and Dynamical Systems, 2008
AbstractLetNbe an integer withN≥2 and letXbe a compact subset of ℝd. If$\mathsf {S}=(S_{1},\ldots ,S_{N})$is a list of contracting similaritiesSi:X→X, then we will write$K_{\mathsf {S}}$for the self-similar set associated with$\mathsf {S}$, and we will writeMfor the family of all lists$\mathsf {S}$satisfying the strong separation condition.
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HAUSDORFF AND PACKING MEASURE OF SETS OF GENERIC POINTS: A ZERO-INFINITY LAW

Journal of the London Mathematical Society, 2004
The main result of the paper consists of a fine analysis of the size of the set \(G_\mu\) of generic points (i.e., satisfying the Birkhoff ergodic theorem for all continuous functions) of an invariant measure \(\mu\) defined on a symbolic space. The symbolic space is endowed with a metric defined via a Gibbs measure \(\mu_\phi\) associated to a ...
Ma, Jihua, Wen, Zhiying
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Exact Hausdorff and packing measures of Cantor sets with overlaps

Ergodic Theory and Dynamical Systems, 2014
Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [
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